I'm running some DC icicle lights through a 6 channel RGB controller at 24vdc (they originally ran at 24vdc too).
The problem is when I run them through the controller, they do not run at the same brightness that they ran at through the original power supply and multifunction controller.

I have attached some pictures below to try to explain better. Is it a possibility that the board is limiting the current flow or something similar? The data sheet for it claims 4amps per channel.

24V is too low so there isn't enough current to light the LEDs. It is the current, not the voltage that lights the LEDs.

To get the LEDs running at the right brightness, you need to calculate the current they should draw and then adjust their input voltage to get them to draw this amount of current. The Big W sets of 200 lights will draw 100mA per channel when your voltage is right. You can modify the light sets to get them running at the same voltage.